Are we precise in which name of God we use outside of Chumash?

Question

In the chumash, there is a distinction between the use of Elokim (midat hadin, the characteristic of justice/law) and the four letter name (midat harachamim, the attribute of mercy).

Rashi on 1:1 reads (as per chabad's text)

God’s creation of the heavens and the earth: But it does not say “of the Lord’s creation of” (i.e., it should say “of the Lord God’s creation of” as below 2:4 “on the day that the Lord God made earth and heaven”) for in the beginning it was His intention to create it with the Divine Standard of Justice, but he perceived that the world would not endure; so He preceded it with the Divine Standard of Mercy, allying it with the Divine Standard of Justice, and that is the reason it is written:“on the day the Lord God made earth and heaven.”

ברא א-להים: ולא אמר ברא ה', שבתחלה עלה במחשבה לבראתו במדת הדין, ראה שאין העולם מתקיים, הקדים מדת רחמים ושתפה למדת הדין, היינו דכתיב (להלן ב ד) ביום עשות ה' א-להים ארץ ושמים:

and 2:4 confirms:

the Lord God: יהוה is His name. אֱלֹהִים [means] that He is the Ruler and Judge over the entire world, and so is this defined everywhere according to its simple meaning: the Lord (יהוה ), Who is Ruler and Judge (אֱלֹהִים).

ה' א-להים: ה' הוא שמו, א-להים שהוא שופט ושליט על כל, וכן פירוש זה בכל מקום לפי פשוטו, ה' שהוא א-להים:

As such, when we explicate pesukim, we can look at the name used and see a level of meaning derived from the attribute invoked, such as here, as this is applied to Yeshayau's text.

Is the same distinction applied to the use of the names in davening? Is it possible to understand particular prayers (like the shmoneh esrei) in light of the name used? When prayers are authored, is there an intentional and specific choice of names (both historically and currently)?

[related somewhat Can't use the name of Gd during spontaneous prayer?

Answer 1

Nefesh HaChaim Shaar Beis Ch. 2:

ומה שמושג אצלינו קצת ואנו מכנים ומתארים כמה תארים ושמות וכנויים ומדות. כמו שמצינום בתורה ובכל מטבע התפלה. כולם הם רק מצד התחברותו יתברך אל העולמות והכחות מעת הבריאה. להעמידים ולהחיותם ולהנהיגם כרצונו ית"ש. (והם אשר קראום בשם השתלשלות הספירות). ולפי כל שנויי פרטי סדרי ההנהגה שמשתלשל ונמשך לזה העולם אם לדין. אם לחסד. אם לרחמים. על ידי כחות העליונים והתמזגותם. משתנים השמות והכנויים והתארים

Paraphrased translation:

The little that we can comprehend and we describe Hashem ... with names ... like we find in the Torah and in the structure of prayer all of them are according to His connection to the world¹ ... and according to the shifts in the details of the order of how the world is run, whether for justice, kindness, or mercy ... the names change.

Nefesh HaChaim includes the names in prayer as names which refer to an attribute of how Hashem relates to the world.

The Avnei Eliyahu commentary to the siddur comments on the meanings of some of the names in context. For example, he explains that הא-ל means above our understanding (see there for the significance).

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¹ as opposed to describing His essence, which cannot be described.